Square Root of 201

√201 at a glance

Exact value

√201

Decimal (10 places)

14.1774468788

Rounded

14.2 · 14.18 · 14.177

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.177447

Prime factorization

3 × 67

Cube root

5.857766

How to simplify √201

The prime factorization of 201 is 3 × 67. Every prime appears only once, so there is no pair to bring outside the radical — √201 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 201, 3 and 67 appear an odd number of times, so √201 is irrational and 14.1774468788 is a rounded value.

Where √201 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √201 lies between 14 and 15. 201 is 5 above 196 and 24 below 225, so the root is closer to 14.

• Straight line between 196 and 225: 14.1724 (0.04% low)
• Tangent from 14, i.e. 14 + 5 ÷ 28: 14.1786 (0.01% high)
• Tangent from 15, i.e. 15 − 24 ÷ 30: 14.2000 (0.16% high)

For √201 the tangent at 14 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 201 is just 5 above 196.

Finding √201 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 201: following the tangent line down to zero simplifies to averaging x with 201 ÷ x.

Start from the nearest whole number, 14 (14² = 196):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √201 = 14.1774468788 to every decimal shown.

√201 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √201 the pattern is [14; 5, 1, 1, 1, 2, 1, 8, 1, 2, 1, 1, 1, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √201 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 201y² = 1. Its smallest solution in positive whole numbers is x = 515,095, y = 36,332.

√201 in geometry and everyday measurements

• A square patio or deck of 201 square feet is about 14.18 ft (14 ft 2 in) on each side, so edging all the way around takes 4 × √201 ≈ 56.7 ft.
• 201 is not a sum of two whole-number squares — the prime factor 3 and 67 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √201 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 14 box, because 1² + 2² + 14² = 201.

Square roots near √201 and related roots

• The cube root of 201 is about 5.857766.
• Four times the radicand doubles the root: √804 = 2 × √201 ≈ 28.354894.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.